- Kolmogorov–Arnold–Moser theorem
The

**Kolmogorov–Arnold–Moser theorem**is a result indynamical system s about the persistence of quasi-periodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in theperturbation theory of classical mechanics.The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by

Andrey Kolmogorov in 1954. This was rigorously proved and extended byVladimir Arnold (in 1963 for analyticHamiltonian system s) andJürgen Moser (in 1962 for smoothtwist map s), and the general result is known as the KAM theorem. The KAM theorem, as it was originally stated, could not be applied to the motions of thesolar system , although Arnold used the methods of KAM to prove the stability ofelliptical orbit s in the planarthree-body problem .The KAM theorem is usually stated in terms of trajectories in

phase space of an integrable Hamiltonian system.The motion of an integrable system is confined to adoughnut -shaped surface, aninvariant torus . Differentinitial condition s of the integrable Hamiltonian system will trace different invariant tori in phase space. Plotting any of the coordinates of an integrable system would show that they are quasi-periodic.The KAM theorem states that if the system is subjected to a weak nonlinear perturbation, some of the invariant tori are deformed and others are destroyed. The ones that survive are those that have “sufficiently irrational” frequencies (this is known as the non-resonance condition). This implies that the motion continues to be quasiperiodic, with the independent periods changed (as a consequence of the non-degeneracy condition). The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true. An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as

**KAM theory**. Notably, it has been extended to non-Hamiltonian systems (starting with Moser), to non-perturbative situations (as in the work ofMichael Herman ) and to systems with fast and slow frequencies (as in the work ofMikhail B. Sevryuk ).The non-resonance and non-degeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom. As the number of dimensions of the system increases, the volume occupied by the tori decreases.

Those KAM tori that are not destroyed by perturbation become invariant

Cantor set s, named "Cantori" byIan C. Percival in 1979.As the perturbation increases and the smooth curves disintegrate we move from KAM theory to

Aubry-Mather theory which requires less stringent hypotheses and works with the Cantor-like sets.**See also***

Arnold diffusion

*Nekhoroshev estimates

*Ergodic theory **References*** cite journal

author = Jürgen Pöschel

title = A lecture on the classical KAM-theorem

journal = Proceedings of Symposia in Pure Mathematics (AMS)

volume = 69 | year = 2001 | pages = 707–732

url = http://www.poschel.de/pbl/kam-1.pdf ("links to PDF file")* Rafael de la Llave (2001) "A tutorial on KAM theory". [

*http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=01-29 Online copy*]*

* [*http://www.math.rug.nl/~broer/pdf/kolmo100.pdf KAM theory: the legacy of Kolmogorov’s 1954 paper*]

* [*http://www.math.gatech.edu/~viveros/Thesis/twistmaps.pdf Early Aubry-Mather theory*]

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